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2007 Apr

2007 April 16

Notation. \(\mathbb C\) is the set of complex numbers, \(D = \{z\in \mathbb C: |z|<1\}\), and if \(G\subset \mathbb C\) is an open set, then \(H(G)\) is the set of holomorphic functions on \(G\).

  1. Give the Laurent series expansion of \(\frac{1}{z(z-1)}\) in the region \(A \equiv \{z\in \mathbb C: 2< |z+2| < 3\}\).

  2. Suppose for all \(n\in \mathbb N\) that the function \(f_n\) is holomorphic in \(D\) and satisfies \(|f_n(z)|<1\) for all \(z \in D\). Also suppose that \(\lim_{n\to \infty} \mathrm{Im}\ f_n(x) = 0\) for all \(x\in (-1,0)\).\ (a) Prove: \(\lim_{n\to \infty} \mathrm{Im}\ f_n(1/2) = 0\).\ (b) Give a complete statement of the convergence theorem that you use in part (a).

  3. Use the residue theorem to evaluate \(\int_{-\infty}^{\infty} \frac{1}{1+x^4}\ dx\).

  4. Present a function \(f\) that has all of the following properties:\ (i) \(f\) is one-to-one and holomorphic on \(D\).\ (ii) \(\{f(z): z\in D\} = \{w \in \mathbb C: \mathrm{Re}\ w > 0, \ \mathrm{Im}\ w > 0\}\).\ (iii) \(f(0) = 1+i\).

  5. (a) Prove: If \(f: D \rightarrow D\) is holomorphic and \(f(1/2) = 0\), then \(|f(0)| \leq 1/2\).\ (b) Give a complete statement of the maximum modulus theorem that you use in part (a).

  6. Prove: If \(G\) is a connected open subset of \(\mathbb C\), any two points of \(G\) can be connected by a parametric curve in \(G\).